© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11a4 K11a6

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X10,6,11,5 X14,7,15,8 X2,9,3,10 X20,12,21,11 X16,14,17,13 X6,15,7,16 X22,17,1,18 X12,20,13,19 X18,21,19,22

Gauss Code:

{1, -5, 2, -1, 3, -8, 4, -2, 5, -3, 6, -10, 7, -4, 8, -7, 9, -11, 10, -6, 11, -9}

DT (Dowker-Thistlethwaite) Code:

4 8 10 14 2 20 16 6 22 12 18

Alexander Polynomial:

- t-3 + 9t-2 - 30t-1 + 45 - 30t + 9t2 - t3

Conway Polynomial:

1 - 3z2 + 3z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{125, 0}

Jones Polynomial:

q-6 - 3q-5 + 7q-4 - 12q-3 + 17q-2 - 20q-1 + 20 - 18q + 14q2 - 8q3 + 4q4 - q5

Other knots (up to mirrors) with the same Jones Polynomial:

{K11a112, ...}

A2 (sl(3)) Invariant:

q-20 + q-18 - 2q-16 + q-14 + q-12 - 4q-10 + 4q-8 - q-4 + 2q-2 - 4 + 3q2 - 3q4 + q6 + 4q8 - 3q10 + 2q12 + q14 - q16

HOMFLY-PT Polynomial:

- a-4z2 + 2a-2 + 2a-2z2 + 2a-2z4 - 3 - 5z2 - 2z4 - z6 + 3a2 + 4a2z2 + 3a2z4 - 2a4 - 3a4z2 + a6

Kauffman Polynomial:

- a-5z3 + a-5z5 + 3a-4z2 - 6a-4z4 + 4a-4z6 - a-3z + 5a-3z3 - 10a-3z5 + 7a-3z7 - 2a-2 + 8a-2z2 - 8a-2z4 - 4a-2z6 + 7a-2z8 - 4a-1z + 16a-1z3 - 24a-1z5 + 8a-1z7 + 4a-1z9 - 3 + 9z2 - 20z6 + 13z8 + z10 - 4az + 18az3 - 23az5 + az7 + 7az9 - 3a2 + 7a2z2 + 3a2z4 - 19a2z6 + 10a2z8 + a2z10 - 3a3z + 15a3z3 - 18a3z5 + 3a3z7 + 3a3z9 - 2a4 + 6a4z2 - 2a4z4 - 6a4z6 + 4a4z8 - 2a5z + 7a5z3 - 8a5z5 + 3a5z7 - a6 + 3a6z2 - 3a6z4 + a6z6

V2 and V3, the type 2 and 3 Vassiliev invariants:

{-3, 2}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=0 is the signature of 115. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 11                       1 j = 9                     3   j = 7                   5 1   j = 5                 9 3     j = 3               9 5       j = 1             11 9         j = -1           10 10           j = -3         7 10             j = -5       5 10               j = -7     2 7                 j = -9   1 5                   j = -11   2                     j = -13 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Show[DrawMorseLink[Knot[11, Alternating, 5]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 5]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[14, 7, 15, 8], > X[2, 9, 3, 10], X[20, 12, 21, 11], X[16, 14, 17, 13], X[6, 15, 7, 16], > X[22, 17, 1, 18], X[12, 20, 13, 19], X[18, 21, 19, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 5]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -8, 4, -2, 5, -3, 6, -10, 7, -4, 8, -7, 9, -11, 10, > -6, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 5]]
Out[5]=	DTCode[4, 8, 10, 14, 2, 20, 16, 6, 22, 12, 18]
In[6]:=	alex = Alexander[Knot[11, Alternating, 5]][t]
Out[6]=	-3 9 30 2 3 45 - t + -- - -- - 30 t + 9 t - t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 5]][z]
Out[7]=	2 4 6 1 - 3 z + 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 5]}
In[9]:=	{KnotDet[Knot[11, Alternating, 5]], KnotSignature[Knot[11, Alternating, 5]]}
Out[9]=	{125, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 5]][q]
Out[10]=	-6 3 7 12 17 20 2 3 4 5 20 + q - -- + -- - -- + -- - -- - 18 q + 14 q - 8 q + 4 q - q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 5], Knot[11, Alternating, 112]}
In[12]:=	A2Invariant[Knot[11, Alternating, 5]][q]
Out[12]=	-20 -18 2 -14 -12 4 4 -4 2 2 4 6 -4 + q + q - --- + q + q - --- + -- - q + -- + 3 q - 3 q + q + 16 10 8 2 q q q q 8 10 12 14 16 > 4 q - 3 q + 2 q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 5]][a, z]
Out[13]=	2 2 2 2 4 6 2 z 2 z 2 2 4 2 4 -3 + -- + 3 a - 2 a + a - 5 z - -- + ---- + 4 a z - 3 a z - 2 z + 2 4 2 a a a 4 2 z 2 4 6 > ---- + 3 a z - z 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 5]][a, z]
Out[14]=	2 2 2 4 6 z 4 z 3 5 2 3 z -3 - -- - 3 a - 2 a - a - -- - --- - 4 a z - 3 a z - 2 a z + 9 z + ---- + 2 3 a 4 a a a 2 3 3 3 8 z 2 2 4 2 6 2 z 5 z 16 z 3 > ---- + 7 a z + 6 a z + 3 a z - -- + ---- + ----- + 18 a z + 2 5 3 a a a a 4 4 5 3 3 5 3 6 z 8 z 2 4 4 4 6 4 z > 15 a z + 7 a z - ---- - ---- + 3 a z - 2 a z - 3 a z + -- - 4 2 5 a a a 5 5 6 6 10 z 24 z 5 3 5 5 5 6 4 z 4 z > ----- - ----- - 23 a z - 18 a z - 8 a z - 20 z + ---- - ---- - 3 a 4 2 a a a 7 7 2 6 4 6 6 6 7 z 8 z 7 3 7 5 7 > 19 a z - 6 a z + a z + ---- + ---- + a z + 3 a z + 3 a z + 3 a a 8 9 8 7 z 2 8 4 8 4 z 9 3 9 10 2 10 > 13 z + ---- + 10 a z + 4 a z + ---- + 7 a z + 3 a z + z + a z 2 a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 5]], Vassiliev[3][Knot[11, Alternating, 5]]}
Out[15]=	{-3, 2}
In[16]:=	Kh[Knot[11, Alternating, 5]][q, t]
Out[16]=	10 1 2 1 5 2 7 5 10 -- + 11 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 7 10 10 3 3 2 5 2 5 3 > ----- + ---- + --- + 9 q t + 9 q t + 5 q t + 9 q t + 3 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a5

K11a4 K11a6